from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,0,13,5]))
pari: [g,chi] = znchar(Mod(1217,2600))
Basic properties
| Modulus: | \(2600\) | |
| Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{325}(242,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2600.em
\(\chi_{2600}(177,\cdot)\) \(\chi_{2600}(473,\cdot)\) \(\chi_{2600}(697,\cdot)\) \(\chi_{2600}(1217,\cdot)\) \(\chi_{2600}(1513,\cdot)\) \(\chi_{2600}(1737,\cdot)\) \(\chi_{2600}(2033,\cdot)\) \(\chi_{2600}(2553,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.148970555205860748537816107273101806640625.1 |
Values on generators
\((1951,1301,1977,1601)\) → \((1,1,e\left(\frac{13}{20}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2600 }(1217, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)